Uniqueness of enhancements for derived and geometric categories

TL;DR

This paper proves that various derived categories associated with abelian categories and schemes have unique enhancements, ensuring a canonical structure for their derived categories across different contexts.

Contribution

It establishes the uniqueness of enhancements for multiple types of derived categories, including unbounded, bounded, and categories related to schemes, extending previous results.

Findings

Uniqueness of enhancements for unbounded, bounded, and bounded above/below derived categories.

Uniqueness of enhancements for unseparated and left completed derived categories of Grothendieck abelian categories.

Uniqueness of enhancements for derived categories of complexes with quasi-coherent cohomology and perfect complexes on certain schemes.

Abstract

We prove that the derived categories of abelian categories have unique enhancements -- all of them, the unbounded, bounded, bounded above and bounded below derived categories. The unseparated and left completed derived categories of a Grothendieck abelian category are also shown to have unique enhancements. Finally we show that the derived category of complexes with quasi-coherent cohomology and the category of perfect complexes have unique enhancements for quasi-compact and quasi-separated schemes.